\( % Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}} \)

Definition of a sphere

A sphere in \(\mathbb{R}^3\) is defined by point vector \(\mathbf{c} \in \mathbb{R}^3\) and a scalar \(r \in \mathbb{R}, r \geq 0\). \(\mathbf{c}\) is called the center of the sphere, \(r\) is called the radius of the sphere. The sphere is the set of all point with a distance to \(\mathbf{c}\) smaller than or equal to the radius \(r\) i.e. \( \left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; |\mathbf{p} - \mathbf{c}| \leq r \right\} \).